A Note on the Eilenberg-Mac Lane Isomorphism for Quadratic Forms
César Galindo
TL;DR
The work provides an elementary, constructive proof of the Eilenberg-Mac Lane isomorphism between $H^3_{2-ab}(G,M)$ and $Quad(G,M)$ by reducing the problem to cyclic components via direct-sum decompositions and extending from finitely generated to general abelian groups through filtered colimits. It gives explicit constructions of the correspondence, characterizes when a quadratic form is a trace of a bilinear form (in terms of 2-primary torsion), and connects the algebraic framework to homotopy theory and categorical groups, including explicit formulas and normal forms. These results clarify when quadratic forms are representable as traces and illuminate the role of 2-torsion in the structure of 2-type models. The combination of concrete cocycle data and categorical/topological connections enhances the utility of the EMtrace isomorphism in applications to fusion categories and homotopy-theoretic models.
Abstract
We give an elementary proof of the Eilenberg-Mac Lane trace isomorphism between the third 2-abelian cohomology group and quadratic forms. Our approach yields explicit constructions and we characterize when quadratic forms can be expressed as traces of bilinear forms for arbitrary coefficient groups.